Discrete mathematics - tree problem

Question

Prove that a connected graph G = ( V, E ) is a tree, if | V | = 6 and |{ v ∈ V | deg( v ) = 1 }| = 4 and |{ v ∈ V | deg( v ) = 3 }| = 2

Answer 1

Hint. A proof could have this overall structure:

Either the two nodes of degree 3 are neighbors, or they are not.

Case 1. If they are neighbors, then ... (bla bla ba) ... and therefore the graph is a tree.

Case 2. If they are not neighbors, then ... (bla bla bla) ... which is a contradiction.

Don't hope to find a slick generic argument that you can use to prove tree-ness given a degree sequence. It only works at all in this particular case because the tree happens to be so small.